The Witt groups of extended quadratic forms over Z

Abstract: We recall the notion of a quadratic form parameter $Q$ over the integers and of extended quadratic forms with values in $Q$, which we call $Q$-forms. Certain form parameters $Q$ appeared in Wall's work on the classification of almost closed $(q{-}1)$-connected $2q$-manifolds via $Q$-forms. The algebraic theory of general form parameters and extended quadratic forms has been studied in a variety of far more general settings by authors including Bak, Baues, Ranicki and Schlichting. In this paper we classify all quadratic form parameters $Q$ over the integers, determine the category of quadratic form parameters $\mathbf{FP}$ and compute the Witt group functor, [ W_0 \colon \mathbf{FP} \to \mathbf{Ab}, \quad Q \mapsto W_0(Q),] where $\mathbf{Ab}$ is the category of finitely generated abelian groups and $W_0(Q)$ is the Witt group of nonsingular $Q$-forms.